étale morphisms

Definition

Étale morphisms are a special type of morphism in algebraic geometry that generalize the notion of smoothness. They can be thought of as morphisms that locally resemble a projection from a product space, allowing for nice properties like being flat and unramified. This concept plays a crucial role in understanding the structure of schemes and their relationships, particularly in the study of idele class groups and comparison theorems.

5 Must Know Facts For Your Next Test

1. Étale morphisms are key in establishing the notion of étale cohomology, which provides tools for computing cohomology groups in algebraic geometry.
2. They are locally of finite presentation, meaning they can be represented by finitely many generators and relations in a suitable local ring.
3. In the context of idele class groups, étale morphisms allow us to study local-global principles, providing insights into how local data influences global properties.
4. Étale morphisms are crucial for proving results such as the existence of local sections in étale sheaves, which aids in constructing various sheaf-theoretic objects.
5. They also connect to comparison theorems, as étale morphisms help establish relationships between different types of cohomological invariants across various fields.

Review Questions

• How do étale morphisms relate to the concept of smoothness in algebraic geometry?
  • Étale morphisms generalize the idea of smooth morphisms by ensuring that they locally resemble projections from products. This means that while smoothness is often tied to differentiability and tangent spaces, étale morphisms focus on maintaining discrete and nice behaviors in fibers. They capture essential aspects of geometry while relaxing some strict conditions found in purely smooth contexts.
• Discuss the significance of étale morphisms in relation to idele class groups and their properties.
  • Étale morphisms play a vital role in understanding idele class groups by enabling the study of local-to-global principles. Through these morphisms, we can analyze how local data at various places contributes to global characteristics of algebraic structures. The behavior of étale morphisms helps reveal information about the class group structure when considering valuations and their corresponding ideles.
• Evaluate the impact of étale morphisms on the development and application of comparison theorems in algebraic geometry.
  • Étale morphisms have a profound impact on comparison theorems by providing a framework to relate different cohomological invariants across fields. Their properties ensure that one can translate results obtained in one context into another effectively. This allows mathematicians to establish deep connections between various areas of study within algebraic geometry, thereby enhancing our understanding of the interplay between geometry and arithmetic through cohomological techniques.

Related terms

Smooth Morphisms: Morphisms between schemes that locally look like a smooth map between manifolds, which ensures certain nice geometrical properties.

Flat Morphisms: Morphisms that satisfy a specific condition on how fibers behave, allowing for the preservation of dimension in a suitable sense.

Unramified Morphisms: Morphisms where the fibers are discrete and have no infinitesimal structure, ensuring that points do not collide under the morphism.